Herman Feshbach, “The Optical Model” - LNS46 Symposium: On the Matter of Particles
PRESENTER: Most people are back, and hope that the sound of the discussion will bring the rest. Couldn't give me more pleasure to do this than to introduce Herman Feshbach than anything else. Herman's been my teacher-- let's say, since 1950, and my collaborator since 1960. As I was saying earlier, he and Viki were the reason that we started the Center for Theoretical Physics and the inspiration for it. And he was the first director of it. We all know who he is, so I'm not going to say any more. Herman.
FESHBACH: Are we on? Yep. Thank you, Arthur. I can't help-- I can't resist but make the following remark at the end of the talks this afternoon. As you all know, economists have proven that it's difficult to predict the future. And we also now know that historians and field theorists find it difficult to predict the past.
When in the good old early days, before the war, there was no theoretical nuclear physics at MIT. There were a number of experimentalists, Robley Evans has been mentioned. Van de Graaff. Bill Buckner and Stan Livingston, who built the cyclotron. That was mentioned earlier today. But there was the ostensible, or the, well, the nuclear theorist was presumably Phil [? Morris. ?] But he had given up nuclear physics. He had done a number of papers with Fisk, Jim Fisk, and Leonard Schiff, but then got interested in acoustics.
So for me, the arrival of Weisskopf, Zacharias, and Rossi was an absolutely great event, the greatest event in my career. Of course, Viki came with a couple of post-docs, Wayne Bowers and Ted Welton. And we got a great deal of excitement and interest in those days. Viki, of course, was at the forefront of the field, and so for the first time, as it were, I was at the forefront of the field. But beyond that, there was Viki as a role model.
I wrote this 20 years ago almost now, when Viki retired. Let me just quote it again.
"His was an unquenchable desire to understand the essential physical elements involved in a phenomenon. To strip away the complexities of a detailed explanation and to make visible the underlying ideas and concepts. It's an attitude which we at MIT particularly have come to appreciate and emulate."
Let me also, while I'm still reminiscing-- we were told today about the halcyon days right after World War II, where nuclear physicists were, how shall I say it, extremely popular. And I have two stories in that regard. One is the Shatto Island story that Viki told you but let me tell you at the beginning of it.
Viki and I and Julian Schwinger rode down on the train from Boston to New York discussing, of course, the new data from Columbia. And we then went up to then the headquarters of the American Physical Society, I think. Maybe it's the American Institute of Physics. I can't quite recall. But anyway, we waited around and then a rickety old bus arrived. We got on it. We crossed the bridge into Queens then from Queens into Long Island at which point we were met by an Arab motorcycle policeman. And the bus drove all the way to Greenpoint, Greenport at the end of the island. All 120 miles or so escorted by motorcycle police. So we were, as they should say, very highly regarded. At Greenport we were given a big, luscious dinner.
Now the other story is a few years later. Bernie Feld had put me down as a possible consultant to a company called Nuclear Development Associates and the AEC decided, Viki was also there, decided to do a security study. So one day I was told to come down-- it was actually rather inconvenient but I, of course agreed --to New York to Columbus Circle, where I was to be-- where I was to be asked questions. I had difficulty getting down. I had started out good and early but the DC-4s at American Airlines that were flying in those days failed, two of them in a row. So I arrived just in time to come into this meeting. Now here I am just-- Let's see, was I an associate professor at the time? I think I was. --but after all, 32 years old or something and the room was filled with people I didn't know, of course. I was all alone. And the chief of the security system for the AEC was there. Admiral [? Cochran ?] had come up from Washington just for this purpose. And I was asked whether I would mind if they would make a transcript of what I said. And I had been forewarned by Viki and so I said, No. I wouldn't mind if they took it down but I'd want a copy. Then the first question came and [INAUDIBLE] asked me, You are a member of the Executive Committee of the Cambridge Association of Scientists. Now that was a forerunner of the FAS. I said, Yes. I was. Question, Have you seen any communist infiltration in that board? The transcript says, Feshbach laughter. And after a while, I found out what was going on. Namely, Wendell Furry was on that board and this dangerous communist would be endangering, of course, our security. The fact is I think, two sides of our popularity.
Now what I'm going to do is narrow my discussion to the discussion of reactions. It's an interesting story in a way because although they are very random influences the result is quite coherent. And so I'd like to make sure we get those random things that did happen. Now Viki came in with, of course, the reputation of being the expert on nuclear reactions. He had invented the evaporation model in '37 and then there was a very seminal paper with Ewing in, I guess, '39 and this is used to this day in that domain of which evaporation theory is valid. We added a little bit on to it later when we discussed or added to it the conditions of conservation of parity in angular momentum. So after he arrived we got to working once more on reactions and, of course, the big thing was resonances. And so with [? Peasley ?] we worked a paper on the boundary condition model which had the great merit of being simple and easy to understand. And in fact at the same time Wigner worked on a similar model but, of course, made it a much more elaborate by making it multichannel whereas we were single channel.
But the big breakthrough came with [INAUDIBLE] data on the total cross-sections for nuclei interacting with the neutrons. I'll show you that curve shortly. And I remember sitting with Hans Beta at a breakfast somewhere, some meeting and we were discussing the day that we both were puzzled as to how that could come out of all the resonances we knew that existed in that neutron nucleus interaction. Well, what developed of course was the paper with Porter and Weisskopf on what we call the cloudy crystal ball model, that's now called the optical model. Which, I really dislike as a term because it really is referring to things which are valid at a much higher energy. This is a low energy story. And Dave Saxon, also worked at that time. And there's a Woods-Saxon potential which everybody knows about. What we got, of course, was a very good--
Now where is that gadget? Ah, there it is.
--a very good match with the data. This is actually the theory. This axis is the energy relayed as the total cross-section in units of the area of the sphere and here is the radius. And you see that this represents a very smooth, slowly energy dependent, except, you know, you get a one over V layer at the beginning here but once you get beyond that it's just a threshold effect. These things vary rather slowly with energy and, actually, also with radius, which is the same as saying with mass number. But that's not the important thing. We first did this with-- Well let me just say that's not the important thing. What the important thing was is in the appendix to that paper.
Viki showed that you could get an understanding of these phenomena by energy-averaging the resonance. In other words, you could start off with a resonance formulation of the cross-sections and then by energy averaging you would develop the optical model. And this is a concept which is of great importance and which we refer to over and over again. Later on he developed it further with Francis Friedman in an article and a Bohr Festival book and showed that the energy averaging had the following effect; it separated the prompt part of the reaction from the delayed part of the reaction. It did not include the light part of the reaction which would be due to, let's say, particles getting caught in the nucleus and remaining there because they're in a resonant state, for example. So the prompt part was described by the optical potential and that's a phenomenon we want to refer to over and over again. An idea we refer to over and over again.
At the same time there were other prompt processes going on. Namely, the so-called direct process, like deuterons in- protons-out, inelastic scattering, and so on. And, for example that was I think Dave [? Peasley's ?] thesis. And so there was a dilemma which was set up if you wish, or a paradox, or a problem-- I should perhaps call it a problem-- in which one would try to develop a formalism which at the same time explained the direct reactions, the prompt reactions if you wish, explain the resonances and by energy averaging gave you the optical model. Now this is a problem which I didn't attack frontally. What happened was that Viki one day he said to me, you know there ought to be a relation like the Kramer's-Kronig Relation in optics. Remember that's the relation between the real and imaginary part of the index of refraction. There ought to be such a relation for nuclear physics, for the passage of say of a neutron through nuclear matter. And that, in fact, led to a theory or formalism which did all the things I just asked for and more.
I must tell you that I got the idea for how to handle that situation sitting in an airplane behind [INAUDIBLE]. So you see, I had great influences working on me. Not only were we able to explain the three things I just mentioned namely, the direct reactions, the optical model and the performance of the appropriate energy averaging, which by the way Michel Baringe made a very helpful remark but we also were able to include the Pauli Principle. The Pauli Principle, of course, has to be there when you have an incident particle, which is a nucleon or composed of nucleons incident on a nucleus. Later we developed these particular papers were for protons and neutrons incident on nuclei. Later we developed extended the formulism, if you wish, for particle transfer reactions in which, let's say you have a proton coming in or a deuteron coming out or heavy ion coming in another heavy ion coming out. So those all can be handled and the Pauli Principle included.
But to return to the main thing that prompted and delayed. Now what are the orders of magnitude of these things? Well, if you take a look at a neutron resonance of a few volts, electron volts, the lifetime is of the order of 10 to the minus 22 seconds, I'm sorry, 10 to the minus 15 seconds. On the other hand, the structure that you saw and see in this thing is of the order of a few meV or more. Now that gives you a lifetime of the time constant if you wish, of the order of 10 to the minus 22, 10 to the minus 23 seconds. So you have an enormous range between 10 to the minus 15th and 10 to the minus 23. And the question is, isn't there something in between? So of course, again, instead of asking the question directly or trying to answer it directly, we looked at some phenomena. You know, without experimentalists we'd be all dead.
And the phenomenon we looked at, I looked at with Barry Bloch was the phenomenon shown here. This is the so-called strength function which is an average of the width of the resonances divided by the energy distance between them averaged as a function of atomic weight. And this solid line was the Cloudy Crystal Ball Model prediction. Now don't worry about this stuff here. That got understood in the terms of the fact that these nuclei were deformed. What interested us was these very low values here which were very far away from the prediction. And the predictions were very good in here but again, there was a bunch here where one might be concerned. But we really worried about this arena there. And the explanation turned out to be that if you send a neutron in, let's say hits a nucleus, it'll hit a nucleon and nucleus, it will get de-excited. The nucleon and nucleus will get excited and on that basis you create a two-particle one-hole state. Two particles excited, one hole where one of the particles was.
Now we thought of this as a doorway state, in the sense that in order to get to the very complicated compound nuclear wave functions one had to pass through that state in order to develop the complexity. And so this phenomenon here here of these low values was an indication that the density of the two-particle one-hole states was low. Barry Block actually calculated that on a model. These points which are empty are his theoretical values. And here are the experimental values of [? fill points. ?] So that was the introduction of doorway states or in French, état de porte. And the thing that naturally came up now was, well, this was an average. So it's an average over perhaps many doorway's states.
There are doorway states which are isolated, that stand out by themselves. And actually, there were at the time three different varieties. There was the giant electric dipole resonance. There was the isobar analog resonance. There was the isomeric fission structure. And so came a question, how do you develop and carry this theory further so that one can describe not only the averages but also the isolated doorway-state resonance of which I gave you some examples. And this was worked out in the paper with Kerman and Dick Lemmer. And I needn't elaborate too much on it except for the following.
I'd like to show you this very nice curve. If you look at this up here is molybdenum PP. Notice the scale. This is 5.2 to 5.4. It's a high-resolution experiment and you get these characteristic fluctuations which occur in nuclear reactions which you can see if you make a good resolution experiment. If you wish, it's a noise spectrum. Now if you take a poorer resolution experiment, you don't do as good a job. In other words, do an average over this whole thing. Look what comes out. That's the bottom curve. Beautiful, beautiful scattering resonance, proving that our fundamental ideas on this matter were correct.
Now the next step, you might think is obvious. Namely, again let us see if we can get structure which is now in between the doorway-state structure, the giant resonance structure and the compound nuclear structure. But again, that's not the way it happened. I went to a meeting in Plitvice Lakes run by Nicholas [? Sindro. ?] That's a very beautiful part of what is now Croatia and I hope it's still there. It's an area with a dozen lakes several waterfalls of varying heights and so forth, a great place. And I, my job was to summarize this conference which was on among other things, what was known as pre-equilibrium reactions. I won't try to define that here, now. And my summary said essentially, hey, this is great stuff but it ain't quantum mechanics and thereby presented myself with a challenge to produce the quantum mechanics.
Well that year, that spring, that was in the summer so the next spring, I had to give invited talk somewhere and I had to invent the theory. Which I did. Which is now called Statistical Multi-compound Reactions. And when I got back to Cambridge I found out that Steve Koonin and Arthur Kerman had invented the same theory. So of course, we got together and developed results. You have to show that there was a reason for doing this beside my adventure in Plitvice. Let me present such a reasoning so you'll see what was going on.
Where are we? I keep losing this thing.
What we have here is a 51-Vanadium PN reaction to chromium. The energy of the proton is 20 to meV. The angle is 144 degrees. It's an experiment done Livermore by Grimes and Anderson and others. And these dark boxes here, the dark points are experiment and this is the prediction of the Weisskopf-Ewing theory, these boxes here. So you see by the time the--
This is the excitation energy of the residual nucleus. So this is low-energy neutrons, high-energy neutrons. So as we got to high-energy neutrons the statistical theory failed miserably to give the experimental results, by orders of magnitude. So that's, of course, a stimulant to look at it. And it was explained by Grimes et al. As a result of the presence of doorway states and the reactions which doorway states were involved. In other words, there were many doorway states and you averaged over them. In that way, you've got the result, which I'll come back to. Well that looked very good and in fact I presented it for Steve and Arthur, myself at the Munich Conference-- an international conference in nuclear physics. And we got a very heated letter from Laura [INAUDIBLE], a very great Italian nuclear physicist. She said, you can't be right. Well, why can't we be right? Well, you predict that the cross-sections, the angle of distributions are symmetric around 90 degrees and I can show you reaction after reaction where they're peaked in a forward direction. Well, that summer I met with Arthur Kerman at Los Alamos and we discussed Laura's problems and we came off with a solution. And I worked the rest of it out on the plane going home, and that became a statistical multi-step direct reaction.
And let me spend a few minutes telling you about that because it's the answer to all reaction problems, if you wish. You take a typical spectrum, let's say a PN reaction, if you wish, this is the energy of neutron, this is the double differential cross-section, this is the spectrum. In this general arena or area, low energy neutrons, you find Viki's evaporation, spherical-angle distribution, very rapid energy dependence-- lots and lots and lots of fluctuations which became known as the Ericson Fluctuations. On this end, with high energy neutrons in other words, not much energy lost to the nucleus, you have the direct reactions which pick up particular levels. And those are the very slow energy dependents, as I pointed out and they're very strongly forward-peaked.
So we have the whole business to explain between here and here. Now what is pretty obvious from the beginning is that interaction time is short here and long there. So this whole energy axis maps roughly onto interaction time. But how do you make the interaction time long? You make it long by developing complicated states. That takes time to do. So not only does energy map into interaction time, it maps as well into complexity. And let me give a diagrammatic picture of that.
How do we get the complexity? Well, we start out an initial state and we develop in one of these a daisy chain in which these states are more and more complicated. And I'll give you an example of that so that you know what I'm talking about. Suppose for example, we have an incident nucleon. It interacts and falls into the well with the other nucleons exciting this one. Of course, here you see you have the two-particle one-hole state. So this is the doorway state. And then you go on with further interactions and you'll get three-particle-- this is wrong-- three-particle two-hole states. Isn't that lovely? And so on. So you get more and more complexity. More and more of the wave function, of course, consists of this, plus that, plus that, amplitudes from each of these. So as you get on further and further excitations, you get more and more complex.
Now, the other case, the P space, you start out again up here and you excite one of these nucleons up. This drops down but doesn't get captured. So this, if you wish, is a free particle plus a one-particle, one-hole state. And that's an example of a complexity.
Now the theory here is developed on the basis of two approximations. One is the so-called Chaining Hypothesis which I'll illustrate back here. The Chaining Hypothesis is essentially a way of saying that this process is Markovian. Namely that the interaction doesn't take you from this state of complexity to that state of complexity. You have to go through all the intermediate stages. In other words, you can [INAUDIBLE] the interaction on a wave function in this particular box will take you either this way, this way, or it'll stay there but it won't skip any of the boxes. So that's one hypothesis.
The other hypothesis is the Random Phase which I'll bring up here. The Random Phase says that the various amplitudes we're going to consider, when averaged, will simply be a sum, as indicated, of the diagonal values. There'll be no U nu, U lambda stuff. Now this is important because I haven't said what nu is. If nu, nu of course are the quantum numbers and in the case of two-space, remember where everything is bound, then quantum numbers are simply J and pi. That is the angle of momentum and the parity. And if you take the average and there's no interference between various values of J, or parity, you get of course an angle of distribution which is symmetric about 90. And that's what [INAUDIBLE] complained about. It is symmetric about 90 if you're dealing with statistical multi-step compound reactions. On the other hand, for the P space where at least one particle is in the continuum, you see that not only do you have the J the pi of the residual nucleus but you also have another quantum number, namely this linear momentum. So the whole averaging issue changes and you therefore get a difference between the multi-step compound and the multi-step direct.
Well, I'm obviously not going to work all of this out for you in detail but let me just show you some of the results. You remember I showed you this curve here. Now if you include multi-step compounds and add that on to this curve you get these x's. That's done with one parameter. I'll come back to the parameters later. And as Steve Koonin worked this out as well as the next one which illustrates the method.
This I believe is Tantalum. It's at a large angle. So that it's a multi-step compound. And here is the experimental curve. And here is what you get just from evaporation. Again you see this huge difference. This is what you get when you add in the multi-step compound. The sum is here. At this end, you've got to add in two neutrons and three neutrons which you simply do from evaporation theory, you can get the final answer. There.
Now in the multi-step direct domain, a lot of, many, many reactions have been studied by [INAUDIBLE], [? Vanetti ?] and Peter-- Peter-- Peter, What's your name? I'll remember it later. I have to hand it to Laura [INAUDIBLE] and [? Vanetti ?] because they got a hold of a pre-print of ours which had an infinite number of mistakes in it, typos of various kinds. They were able to decipher it and actually go to work and this is the result. Hodgson, Peter Hodgson.
So what we have here again is U, that's the excitation energy. This is the neutron energy spectrum. This is 30 degrees. Energy is 45 meV. You can see this is the result if we have only one step. So this [INAUDIBLE], the direct, usual direct calculation. And as you see it will fail at low energies. You add in two steps, three steps. These are so far down they don't matter. And your final result is this curve. And it matches pretty well with the data, not in this region because they didn't add in the two and three neutron evaporation things. So bear it in mind that in this case one single collision did very well, at least for the high energy neutrons.
Now if you go to 90 degrees, here is again the result. And you can see that a single step just fails very quickly. You've got to add in two and three steps before you can get this curve here. Now if you go to four steps-- I'm sorry, if you put it at another angle, so now 120 degrees. And now you see one step really is only of value at the very highest energy. Two steps and three steps are needed in order to get this curve. These are semi-log plots. So these deviations are substantial but the general tenor and behavior of the cross-sections are well represented.
Let me do one more of these. This is now the angle of distribution. Up to the time that this theory was developed, really, people depended upon the Direct Reaction Theory and so they didn't do anything about the big angles. Which as you can see, this is the excitation energy of the nucleus. And you can see for both high and low energy. It does quite well. That's rather old, actually but there've been a great many experiments done since then with that kind of treatment. And I'll give you the last one I saw, which is now at 150 meV. Remember the other data was at 45. And we've also done it at lower energies. Well, the one with Grimes, et al. Was at 20.
So here we have nickel, 150 meV. And these are the various energies of the emerging protons. And I'll just point out that here are the experimental points and the fit is here. Not so hotsie-totsie here. But after all, if a theory is worth anything it has to be wrong sometimes. If it can't be wrong, it can't be right. Now here is the single scattering, one step, two steps, three steps, four steps, five steps. And, of course, as you can see as you increase the number of steps you have the gut feeling that this is doing the right thing because in order to get to big angles you should need more steps.
And there's another one of these curves. The authors are down here. They're a group from South Africa. And here's Hodgson and [? Vanetti ?] who did, of course, the work and analyzing. Now I think this is different. 100. 120. 150. And you can see the nature of the fits at each of these energies.
Now, if you're interested in phenomenology and that sort of thing you say, well, how many parameters did you use to fit that data? The answer is at a given energy only one parameter, and independent of the nucleus, and independent of whether it was a statistical multi-step compound or a statistical multi-step direct. And here are the parameters.
This is the coupling potential, v0, right here which gives you the probability, if you wish, the magnitude and the probability of going from one complexity to the next complexity. And it falls along a curve like this. That's a totally reasonable curve. It's the same curve, essentially parallel to the behavior of the central part of optical potential as a function of energy.
So I'll finish up with just two other examples. As you remember, Viki said, is there a Kramers-Kronig relation for nuclear interactions? The answer is yes. And in that early paper '58, 1958, that relationship was developed.
Recently Claude [INAUDIBLE] has looked at this in some detail and I want to show you his result. Remember, you relate the real to the imaginary part. The imaginary part, the optical [INAUDIBLE] essentially, you get looking at reaction rates. And so usually you have pretty good information about those. And then you stick that into the relationship and predict the real part. So he tested it out and here it is. By the way, not just for this case but never mind. Now this is not quite the potential. It's the integral of the potential over space. V represents the real part. W the imaginary part. And as you expect the imaginary part has a threshold and then it's fairly constant. Actually, eventually it grows logarithmically but over this domain this is a representation. So he approximated this by a functional straight line plus a constant. Then you can do the dispersion intervals and this is the prediction. For the solid line, the solid line. For the broken line, the broken line. And these points here are experimental points. So you see, yes, there is a Kramers-Kronig relation, Professor Weisskopf for nuclear matter.
Now I didn't tell you what the answers were like for the multi-step compound, multi-step direct. They're really look good when you see them at the end. They feel right. And for the multi-step direct, what it consisted of is a series of foldings. You take a scattering, the first step, and so you get to the first stage of complexity. Let's say you started out with an initial momentum, k sub [? i, ?] you go to case of one. Then you take another step. That means you go from states with K's of one, to states with K's of two. You do that by folding. Then if you go from K2 to K3 and so forth until you come to end steps, and at the end you want to have the final momentum. So that means you've achieved the final momentum in end steps. And then you sum over n.
Now there is a phenomenon in heavy ions called deep inelastic scattering, in which the two have heavy ions have a nice conversation with each other, but they don't get into bed with each other. They don't they form compound nuclei. They exchange particles, verbs, nouns, adjectives, and so on. But they hang around for quite a long time. But they don't as they say form a compound nucleus.
Now this is just set up for multi-step direct reactions and we looked at that with [INAUDIBLE], and in the limit in which the change in momenta and the change in angular momenta are between steps that are small. That is going from step 1, 2, 3 and 4, each step involves a small change. If you do that then you derive from our general expressions a Fokker-Pont equation in momentum and angular momentum space, and then by ingenious strategies you can integrate that equation. That's a law by the way. It's the approximation that's used. It has the advantage that it is quick and dirty and gives you an answer, and you can do it in a relatively short time.
And this is the kind of result you get. This reproduction didn't come out so well. It's this cross-section. U again is the excitation energy. Theta is the angle. So it's d theta, notice, not sine d theta. It's 40. Argon a 40 and thorium 432, and the energy is 388 meV.
And this is the experimental situation. These are three fits and this is the one we like the best. You can see this part of the structure is well developed. This part of the structure is also well developed. The thing we miss is this piece in here. This, of course, that's experimental error. But as you can see, the theory does a nice job in at least qualitatively looking at this thing.
I'm going to stop here. I could go on for an indefinite period. I haven't mentioned a lot of reaction work that was done at MIT. There was the infamous, or famous, Kerman, McManus, and Thaler description of multiple scattering, which was the basis of the understanding of, let's say, one geV protons on nuclei. There's the time dependent Hawtrey-Fawx which maybe Steve will talk about. There is the boundary conditional model that Earle Lomon has used in discussing pi nucleon scattering. And there's the doorway state formalism that [INAUDIBLE]-- Lenz, [INAUDIBLE], and Yazaki used in discussing pi nucleus scattering. They had a rather ingenious generalization which I wish I had time to discuss with you.
Once this work was done while I was administrating like mad, which I did for about 20 odd years, and the only reason it got done was because I had superb students who did all the work and I had the collaboration with my friends, Viki Weisskopf and Arthur Kerman. And I want to take this opportunity to thank them publicly, acknowledge my debt to them publicly, and I also want to thank the Laboratory for Nuclear Science which made it possible to work that hard. Thank you.